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We use k-Schur functions to get the minimal boundary of the k-boundedpartition poset. This permits to describe the central random walks on affineGrassmannian elements of type A and yields a polynomial expression for theirdrift. We also recover Rietsch's parametriza-tion of totally nonnegativeunitriangular Toeplitz matrices without using quantum cohomology of flagvarieties. All the homeomorphisms we define can moreover be made explicit byusing the combinatorics of k-Schur functions and elementary computations basedon Perron-Frobenius theorem.

Alcove random walks, k-Schur functions and the minimal boundary of the k-bounded partition poset